Figures

Abstract

Changes in synaptic efficacies need to be long-lasting in order to serve as a
substrate for memory. Experimentally, synaptic plasticity exhibits phases
covering the induction of long-term potentiation and depression (LTP/LTD) during
the early phase of synaptic plasticity, the setting of synaptic tags, a trigger
process for protein synthesis, and a slow transition leading to synaptic
consolidation during the late phase of synaptic plasticity. We present a
mathematical model that describes these different phases of synaptic plasticity.
The model explains a large body of experimental data on synaptic tagging and
capture, cross-tagging, and the late phases of LTP and LTD. Moreover, the model
accounts for the dependence of LTP and LTD induction on voltage and presynaptic
stimulation frequency. The stabilization of potentiated synapses during the
transition from early to late LTP occurs by protein synthesis dynamics that are
shared by groups of synapses. The functional consequence of this shared process
is that previously stabilized patterns of strong or weak synapses onto the same
postsynaptic neuron are well protected against later changes induced by LTP/LTD
protocols at individual synapses.

Author Summary

Humans and animals learn by changing the strength of connections between neurons,
a phenomenon called synaptic plasticity. These changes can be induced by rather
short stimuli (lasting sometimes only a few seconds) but should then be stable
for months or years in order to be useful for long-term memory. Experimentalists
have shown that synapses undergo a sequence of steps that transforms the rapid
change during the early phase of synaptic plasticity into a stable memory trace
in the late phase. In this paper we introduce a model with a small number of
equations that can describe the phenomena of induction of synaptic changes
during the early phase of synaptic plasticity, the trigger process for protein
synthesis, and the final stabilization. The model covers a broad range of
experimental phenomena known as tagging experiments and makes testable
predictions. The ability to model the stabilization of synapses is crucial to
understand learning and memory processes in animals and humans and a necessary
ingredient for any large-scale model of the brain.

Funding: This work was partially supported by the European community via th FACETS
project. CC was supported by the Swiss National Science Foundation. The sponsors
did not influence the design or analysis of the study.

Competing interests: The authors have declared that no competing interests exist.

Introduction

Changes in the connection strength between neurons in response to appropriate
stimulation are thought to be the physiological basis for learning and memory
formation [1],[2]. A minimal requirement for proper memory function
is that these changes, once they are induced, persist for a long time. For several
decades, experimentalists have therefore focused on Long-Term Potentiation (LTP) and
Long-Term Depression (LTD) of synapses in hippocampus [3],[4] and cortical areas [5],[6]. LTP
can be induced at groups of synapses by strong ‘tetanic’
high-frequency stimulation of the presynaptic pathway [3] while stimulation at
lower frequency leads to LTD Dudek92. Both LTP and LTD can also be induced at a
single synapse or a small number of synaptic contacts if presynaptic activity is
paired with either a depolarization of the postsynaptic membrane [5],[7] or
tightly timed postsynaptic spikes [8],[9].

While the induction protocol for LTP and LTD is often as short as a few seconds, the
changes in synaptic efficacy persist for much longer [9]. In typical slice
experiments on LTP [and similarly for LTD or Spike-Timing Dependent
Plasticity (STDP)] the persistence of the change is monitored for 30
minutes to 1 hour. Accumulating evidence suggests, however, that after this early
phase of LTP (E-LTP) different biochemical processes set in that are necessary for
the further maintenance of potentiated synapses during the late phase of LTP (L-LTP)
[10],[11]. For an
understanding of the transition from early to late LTP, the concept of
‘synaptic tagging and capture’ has become influential [12],[13].
During induction of the early phase of LTP, each potentiated synapse sets a tag that
marks that it has received a specific afferent signal. A candidate molecule,
involved in the tag signaling LTP induction in apical dendrites of hippocampal
neurons, is the calcium-calmodulin dependent kinase II (CaMKII) [13]. Newly
synthesized plasticity-related proteins are ‘captured’ by the
tagged synapse and transform E-LTP into L-LTP that can be maintained over hours or
days. A candidate protein involved in the maintenance of potentiated hippocampal
synapses is the protein kinase Mζ
(PKMζ) [11],[14].

The stabilization and maintenance of potentiated synapses poses a number of
theoretical challenges. First, on the level of single synapses we must require
synaptic strength to remain stable, despite the fact that AMPA channels in the
postsynaptic membrane are continuously exchanged and recycled [15]–[17]. Thus
the synapse is not ‘frozen’ but part of a dynamic loop. Second,
on the level of neuronal representation in cortical areas, one finds representations
of input features that are stable but at the same time sufficiently plastic to
adjust to new situations [18]. In the theoretical community, this paradox has
been termed the stability-plasticity dilemma in unsupervised learning [19].
Third, humans keep the ability to memorize events during adulthood, but can also
remember earlier episodes years back. However, continued learning of new patterns in
theoretical models of associative memory networks forces the erasure or
‘overwriting’ of old ones, the so-called palimpsest property
[20],[21]. In the context of continued learning,
theoretical arguments show that synaptic plasticity on multiple time scales cannot
prevent, but at most delay the erasure of memories in the presence of ongoing
synaptic activity [22]. This suggests that additional mechanisms are
necessary to further protect existing memories and ‘gate’ the
learning of new ones.

Despite these challenges for the long-term stability of synapses, most classical
models of synaptic plasticity focus on the induction and early phase of LTP or LTD
and completely ignore the question of maintenance. Traditional models of associative
memories separate the learning phase from the retrieval phase [23] and the same holds
for standard models of STDP [24]–[26]. Detailed biophysical
models of LTP and LTD describe calcium dynamics and Calcium/Calmodulin-Dependent
Protein Kinase II (CaMKII) phosphorylation during the induction and early phase of
LTP [27]–[29]. While these models
show that switches built of CaMKII proteins can be stable for years, they do not
address aspects of tagging leading to heterosynaptic interaction during L-LTP and
L-LTD. Moreover, while CaMKII phosphorylation is necessary for induction of LTP and
mediate tags in the apical dendrites of hippocampal CA1 neurons [30], it
is less clear whether it is necessary for its maintenance [31]. On the other hand
protein kinase Mζ is essential for maintenance of some
synapse types [11],[13],[14] but the same
molecule is potentially relevant for induction in others [30].

We wondered whether a simple model that connects the process of LTP induction with
that of maintenance would account for experimental results on tagging and
‘cross-tagging’ [11]–[13],[32]
without specific assumptions about the (partially unknown) molecular pathways
involved in the maintenance process. If so, the model should allow us to discuss
functional consequences that are generic to the tagging hypothesis independent of
the details of a biophysical implementation in the cell. Even though we believe that
the model principles are more general, we focus on synapses from the
Schaffer-Collaterals onto the CA1 neurons in hippocampus as an experimentally
well-studied reference system for synaptic plasticity. Since typical tagging
experiments involve the extracellular stimulation of one or several
groups of synapses (rather than single synapses), our model of
early and late LTP/LTD is developed in the context of a neuron model with hundreds
of synapses. The application of the principles of synaptic consolidation to
experiments inducing E-LTP/E-LTD at single synapses is considered
in the discussion section.

Results

We study a model with a large number of synapses i onto a single
postsynaptic neuron. To be specific, we think of a pyramidal neuron in the CA1 area
of hippocampus. Our model combines features of traditional models for the
induction of potentiation [24]–[26], [33]–[36] with a simple
description of tagging and synthesis of plasticity related proteins that finally
lead to the maintenance of the induced changes. The section is
organized as follows: We first introduce the essential components of the model step
by step (‘Constructing the Model’). We then test the performance
of the model with a set of stimuli typically used to induce long-term changes of
synapses (‘Testing the Model’).

Constructing the Model

Our model contains three elements, Figure 1. The first one sets the tag during the
induction of E-LTP or E-LTD. A tag is indicated by a value
h = 1 for LTP or
l = 1 for LTD. In the absence
of tags we have
h = l = 0.
The second one describes the process that triggers the
synthesis of plasticity related proteins. The final component describes the
up-regulation of a maintenance-related process from a low value
(z = 0) to a high value
(z≈1). The dynamics of this component is
intrinsically bistable and leads to a consolidation of the
previously induced change at the labeled synapses upon interaction with the
protein p (‘protein capture’). The total
change Δw of the synaptic strength reported in
experiments contains contributions [13] of the early
components l and h as well as the late
component z. Since the model describes a sequence of three
steps ‘Tag-Trigger-Consolidation’ we call it in the
following the TagTriC-Model (Figure
1).

Figure 1. The three components of the Tag-Trigger-Consolidation (TagTriC)
model.

(A) A synapse can be in the non-tagged state N, the high state H or the
low state L. A synapse i in H (or L) has a tag
hi = 1
(or li = 1,
respectively). Transitions to a tagged state occur with rates
ρH for potentiation and
ρL for depression. The tag
hi = 1
is indicated by a red flag in both the flow graph and the schematic
drawing below. (B) Synthesis of plasticity related proteins
p (green squares) is triggered if the total number of
set tags is larger than a critical number
Np. If the trigger threshold
Np is not reached, the protein
concentration decays back to zero. (C) The consolidation dynamics can be
visualized as downward motion in a potential surface
E(z). The function
f(z) (shown to the right) is the
derivative of E and characterizes the dynamics
dz/dt = f(z).
If a tag is set at the synapse
(hi = 1)
and protein synthesis has been triggered (p≈1),
the dynamics can be imagined as downward motion into the right well of
the potential E(z). In this case,
z≈1 is the only fixed point of the dynamics
(magenta circle). In the absence of tags
(hi = li = 0,
below) the consolidation variable zi of
synapse i is bistable and approaches (direction of flow
indicated by arrows) stable fixed points at
zi = 0 or
zi = 1
(magenta circles). The steps of synaptic tagging and capture are
indicated immediately below the flow diagram. (D) The tagging rates for
depression (−ρL,(magenta))
and for potentiation ρH (blue) are
shown as a function of the clamped voltage under the assumption that a
presynaptic spike has arrived less than 1 millisecond before. Note that
for depression we plot the negative rate
−ρL rather than
ρL to emphasize the fact
that depression leads to a down-scaling of the synapse. (E) Voltage
dependence of early LTP and LTD. The weight change
Δw/w(0) induced by a
stimulation of 100 synapses at 2 Hz during 50 s while the postsynaptic
voltage is clamped is shown as a function of voltage. The percent change
Δw/w̅ in
simulations (circles) of LTP/LTD induction experiments can be predicted
from a theory (solid line) based on the difference in transition rates
ρH−ρL.
The simulation reflects the voltage dependence seen in experiments [5],[39]. (F,G)
Frequency dependence of early LTP and LTD. Simultaneous stimulation of
100 synapses by 3 trains (separated by 5 min) of 100 pulses at rates
ranging 0.03 to 100 Hz shows LTD at low frequencies and LTP at
frequencies above 30 Hz. (G) If LTP is blocked in the model, LTD (pink
line) occurs up to high frequencies as in experiments [7]. Blue line: LTP with blocked of LTD.

Tag and Induction of LTP/LTD

Results from minimal stimulation protocols which putatively activate only a
single synapse suggest that the induction of LTP is a switch-like process [7],[37]. We therefore
model individual synapses as discrete quantities that can switch, during the
induction of LTP, from an initial ‘non-tagged state’ (N) to
a ‘high state’ (H) with a transition rate
ρH that depends on the induction
protocol. Similarly, induction of LTD moves the synapse from the initial
non-tagged state (N) to a ‘low state’ (L) at a rate
ρL. If synapse i is
in the high state, the synaptic variable hi is equal
to one. If it is in the low state, another local variable
li is set to one. These local variables
hi and li do not
only control the weight of the synapse during E-LTP and E-LTD, but also serve as
‘tags’ for up- or down-regulation of the synapse. Tags reset
to zero stochastically with a rate kh and
kl, respectively. If both tags are zero, the
synapse is in the non-tagged state N. Since the synapse is either up-regulated
OR down-regulated, at most one of the tags can be non-zero (Figure 1A).

The stochastic transitions from the initial state N with
hi = 0 and
li = 0 to
the down-regulated state
li = 1 or an
upregulated state
hi = 1 depend in a
Hebbian manner on presynaptic activity and the state of the postsynaptic neuron.
In the absence of presynaptic activity, the LTD rate
ρL vanishes. Presynaptic activity
combined with a time-averaged membrane potential u̅
above a critical value ϑLTD leads in the
TagTriC model to a LTD transition rate ρL
proportional to
[u̅(t)−ϑLTD].
For a transition from the initial state to the high state, we require in
addition that the momentary membrane potential is above a
second threshold ϑLTP. Hence the
transition rate ρH is proportional to
[u̅(t)−ϑLTD][u−ϑLTP]
whenever these threshold conditions are satisfied; see Methods for details.

Our assumptions regarding the transition rates essentially summarize the
qualitative voltage dependence seen in the Artola-Bröcher-Singer
experiments [5]. Indeed, when 100 synapses in the TagTriC
model are stimulated at low frequency during 50 seconds while the membrane
voltage is kept fixed at different values (Figure 1D), the total weight change summed
across all synapses exhibits LTD at low voltage and LTP at high voltage [38],[39]. As expected,
the resulting weight changes in the simulations of Figure 1E reflect the voltage dependence of
the transition rates in Figure
1D.

Trigger for Protein Synthesis

Previously induced LTP or LTD needs to be consolidated in order to last for more
than one hour. Consolidation requires that protein synthesis is triggered.
Experimental evidence indicates that triggering of protein synthesis needs the
presence of neuromodulators such as dopamine (in the apical CA1 region) or other
modulators (in other regions). In typical tagging experiments, extracellular
stimulation co-stimulates dopaminergic input leading to a phasic dopamine signal
[13],[40]. In our model,
induction of E-LTP or E-LTD through appropriate stimulation protocols changes
the synaptic efficacy and sets tags at the modified synapses, both described by
the variables
hi = 1 or
li = 1.
Protein synthesis in the model is triggered (see methods for details) if the total number of tags Σi(hi+li)
(which indirectly reflects the phasic dopamine signal) reaches a threshold
Np which depends on the level of background
dopamine (and other neuromodulators). More specifically,
Np decreases with the concentration of background
dopamine so that the presence of dopamine facilitates the trigger process [32].

If the trigger criterion is satisfied, the concentration p of
synthesized plasticity related proteins approaches with rate
kp a value close to one. If the number of tags
falls below the threshold Np, the protein
concentration p decays with a time constant
τp back to zero. Further details on
the role of the trigger threshold and its relation to neuromodulators can be
found in the discussion section.

Consolidation and Late LTP

The total weight wi of a synapse i
depends on the present value of the tags hi or
li as well as on its long-term value
zi. The slow variable
zi is a continuous variable with one or two stable
states described by a generic model of bistable switches, that could be
implemented by suitable auto-catalytic processes [16]. While the
concentration p of plasticity related proteins is zero, the
variable zi has two stable states at
zi = 0 and
zi = 1,
respectively. If the protein concentration takes a value of
p≈1, one of the stable states disappears and, depending
on the tag that was set, the long term-value of the synapse can be up- or
down-regulated; see methods and Figure 1C for details.

In order to illustrate the mechanism of induction of L-LTP, let us suppose that
the synapse has been initially close to the state
zi = 0. The
dynamics of the synapse can be imagined as downward motion in a
‘potential’ E. The current stable state of
the synapse is at the bottom of the left well in the potential pictured in Figure 1C. We assume that
during a subsequent LTP induction protocol the synapse has been tagged with
hi = 1 and
that the total number of tags set during the LTP induction protocol surpasses
the trigger threshold Np. If the protein
concentration p approaches one, the potential surface is tilted
so that the synapse now moves towards the remaining minimum at
z≈1. After decay of the tags, p returns
to zero, and we are back to the original potential, but now with the synapse
trapped in the state z = 1. It
can be maintained in this state for a long time, until another strong tagging
event occurs during which the synapse is tagged with
li = 1 as a result
of LTD induction. In this case the potential surface can be tilted towards the
left so that the only equilibrium point is at
z = 0. Since consolidation is
typically studied in animals that are more than 20 days old [13],
we assume that before the beginning of the experiment 30 percent of the synapses
are already in the upregulated state
z = 1 and the remaining 70
percent in the state z = 0;
see also [7]. Because of the bistable dynamics of
consolidation, only synapses that are initially in the upregulated state
z = 1 can undergo L-LTD
and only synapses that start from
z = 0 can undergo L-LTP;
compare [7]. Note, however, that tags for potentiation and
depression can be set independently of the value of z. We may
speculate that the variable z is related to the activity of
PKMζ[11],[14], or to the
self-sustained clustering of AMPA receptors [41], but the exact
biochemical signaling chain is irrelevant for the functional consequences of the
model discussed in the results section. In our model, the bistable dynamics of
the z-variable captures the essence of synaptic persistence
despite molecular turnover [15],[16],[28] and
mobility of AMPA receptors [41].

Tests of the Model

The TagTriC model has been tested on a series of stimulation protocols that
reflect induction of LTP and LTD as well as the consolidation of plasticity
events.

Induction of Synaptic Changes

A typical LTP induction experiment starts with extracellular stimulation of a
bundle of presynaptic fibers (i.e., the Schaffer collaterals leading from CA3 to
CA1) that activate a large number (typically hundreds [13]) of presynaptic
terminals. With an extracellular probe electrode placed close to one of the
postsynaptic neurons, a change in synaptic efficacy is measured via the
amplitude (or initial slope) of the evoked postsynaptic potential, representing
the total response summed across all the stimulated synapses. In our
simulations, we mimic these experiments by simultaneous stimulation of 100
synapses. The state of the postsynaptic neuron is described by the adaptive
exponential integrate-and-fire model [42] and can be
manipulated by current injection.

In a preliminary set of simulation experiments done with presynaptic stimulation
alone (no manipulation of the postsynaptic neuron), the TagTriC model exhibits
LTD or LTP depending on the frequency of the presynaptic stimulation (Figure 1F) in agreement with
experimental results [4],[43]. Moreover, under
the assumption that LTP has been blocked pharmacologically
(ρH = 0
in the model), our model shows LTD even for high stimulation frequencies (Figure 1G). This stems from
the fact that LTD and LTP are represented in the TagTriC model by two
independent pathways (Figure
1A) which are under control condition in competition with each other, but
show up individually if one of the paths is blocked [43]. Together with the
voltage dependence of Figure
1E, the above simulation results indicate that our model of LTP and LTD
induction can account for a range of experiments on excitatory synapses in the
hippocampal CA1 region, in particular, voltage and frequency dependence.

Consolidation of Synaptic Changes

In order to study whether consolidation of synaptic changes in our model follows
the time course seen in experiments, we simulate standard experimental
stimulation protocols [12],[13]. A weak tetanus
consisting of a stimulation of 100 synapses at 100 Hz for 0.2 seconds (21
pulses) leads in our model to the induction of LTP (change by +15
percent) which decays back to baseline over the time course of two hours (Figure 2A). Thus, after the
early phase of LTP the synapses are not consolidated. A stronger stimulus
consisting of stimulating the same group of hundred synapses by 100 pulses at
100 Hz (repeated 3 times every 10 minutes) yields stronger LTP that consolidates
and remains elevated (weight change by 22±5 percent) for as long as
the simulations are continued (more than 10 hours, only the first 5 hours are
shown in Figure 2B). Thus
our model exhibits a transition from early to late LTP if E-LTP is induced by
the strong tetanic stimulation protocol, but not the weak one, consistent with
results in experiments [12],[13]. If, however, the
weak tetanus at a first group of 100 synapses is given 30 minutes before or
after a strong tetanus at a second group of 100 synapses, the synapses in both
the weakly and strongly stimulated groups are consolidated (Figure 2C and 2D). If the weak tetanus in
group one is given 120 minutes after the strong tetanus in group two, then
consolidation of the synapses in the weakly stimulated group does not occur
(Figure 2E). Thus our
model exhibits a time course of heterosynaptic interaction between the two
groups of synapses as reported in classical tagging experiments [12],[13].

(A) A weak tetanus (21 pulses at 100 Hz) applied at a group of 100
synapses at t = 10 min
(arrow) leads to an increased connection weight
(w/w(0), blue line) that decays
back to baseline. (B) A strong tetanus (100 pulses at 100 Hz repeated
three times, arrows) leads to late LTP that is sustained for 5 hours
(black line). (C) If the weak tetanus (blue arrow) in a first group of
synapses is followed thirty minutes later by a strong tetanus (black
arrows) in a second group of synapses, the weights in the first group
(blue line) and the second group (black line) are stabilized above
baseline. (D) Stimulating a group of synapses by a weak tetanus (blue
arrow) 30 minutes after the end of the strong tetanic
stimulation of a second group also leads to stabilization of the weights
in both groups above baseline. (E) If the weak tetanic stimulation
occurs 2 hours after the strong tetanic stimulation of the other group,
only synapses in the strongly stimulated group will be stabilized (black
line), but not those in the weakly stimulated group (blue line). (F)
Fraction of stabilized weights
Δw/w(0) in the weakly
stimulated group measured 10 hours after induction of LTP as a function
of the time difference between the weak stimulation and the end of the
strong tetanic stimulation in the second group. Blue line: normal set of
parameters
(Np = 40).
Black line: protein trigger threshold increased to
Np = 60. In
panels A–E, lines indicate the result averaged over 10
repetitions of the simulation experiments and bars standard deviation.
In panel F, line indicates the result averaged over 100 repetitions. 90
of the 100 individual trials stayed within the bounds indicated by the
error bars.

An advantage of a modeling approach is that we can study the dependence of the
heterosynaptic interaction between the two groups of synapses upon model
parameters. A critical parameter in the model is the trigger threshold
Np that needs to be reached in order to
start protein synthesis (Figure
1B). With our standard choice of parameters, where
Np = 40, we can
plot the consolidated weight change
Δw/w(0) in the weakly stimulated group
(measured 10 hours after the induction) as a function of the time difference
between the stimulation of the group receiving the strong tetanus and that
receiving the weak tetanus. The curve in Figure 2F shows that for a time difference up
to 1 hour there is significant interaction between the two groups of synapses
leading to synaptic consolidation, whereas for time differences beyond 2 hours
this is no longer the case. If the trigger threshold is increased to
Np = 60
(corresponding to less available neuromodulator), then the maximal time
difference that still yields L-LTP in the weakly stimulated group of synapses is
reduced to about 20 minutes (Figure
2F) whereas a reduction of Np yields an
increased time window of interaction (data not shown). If
Np is reduced much further, the weak tetanus alone
will be sufficient to allow a transition from the early to the late phase of
LTP. We speculate that Np could depend on the age of
the animal as well as on the background level of dopamine or other
neuromodulators so as to enable a tuning of the degree of plasticity (see
discussion for details).

LTD and Cross-Tagging

We consider two experimental protocols known to induce LTD—a weak
low-frequency protocol consisting of 900 pulses at 1 Hz and a strong
low-frequency protocol consisting of 900 repetitions at 1 Hz of a short burst of
three pulses at 20 Hz. This strong low-frequency protocol applied to 100 model
synapses leads to a significant level of LTD (reduction of weights to
70±4 percent of initial value) which is consolidated 5 hours later at
a level of 83±3 percent of initial value. If a group of 100 synapses
is stimulated with the weak low-frequency protocol, an early phase of LTD is
induced that is not consolidated but decays over the time course of 3 hours
(Figure 3A and 3B).
However, if the weak low-frequency stimulation occurs after another group of 100
synapses had been stimulated by the strong low-frequency protocol, then the
group that has received the weak stimulation shows consolidated synapses (at
90±2 percent 5 hours after stimulus induction, Figure 3C). Moreover, consolidation of LTD
(at 92±3 percent 5 hours after stimulus induction) in the group of
synapses receiving the weak low-frequency protocol also occurs if it was
stimulated thirty minutes after the stimulation of a second group of synapses by
a strong tetanus, leading to LTP (Figure 3D). Thus, the TagTriC model exhibits cross-tagging
consistent with experiments [11],[32]. In our model,
cross-tagging occurs because the tags for LTP and LTD
(hi and li,
respectively) enter in a symmetric fashion into the trigger criterion for the
synthesis of plasticity-related proteins (see Figure 1 and Methods).

(A) A strong low-frequency stimulus (3 pulses at 20 Hz, repeated 900
times every second) applied to a group of
N = 100 synapses
induces LTD with mean weights (w/w(0))
stabilized at 83±3% of initial value after 5 hours
(black line). (B) A weak low-frequency stimulus (1 pulse repeated 900
times at 1 Hz) induces early LTD, which is not consolidated. (C) If the
weak low-frequency stimulus is applied 30 minutes after a second group
of synapses has received the strong low-frequency protocol, the weights
in both groups (blue, weak stimulus; black, strong stimulus) are
consolidated at values below baseline. (D) Consolidation of LTD in the
group receiving weak low-frequency stimulation (blue line) also happens
if induction occurs 30 minutes after stimulating a second group of
synapses with a strong tetanic protocol (see Figure 2) inducing LTP (black line).
Downward arrows indicated the period of weak (blue arrow) or strong
(black arrow) low-frequency protocols. The black upward arrows indicate
strong tetanic stimulation. Lines show mean results, averaged over 10
repetitions of the simulation experiment. Error bars are standard
deviation.

Model Mechanism for Tagging, Cross-Tagging, and Consolidation

In order to elucidate how the model gives rise to the series of results discussed
in the preceding paragraphs, we have analyzed the evolution of the model
variables during and after induction of LTP (Figure 4). Critical for consolidation is the
synthesis of plasticity related proteins, characterized by the variable
p in the model. Synthesis is only possible while the total
number of tags is above the protein triggering threshold
Np. For the strong tetanic stimulus this
criterion is met for about 90 minutes (shaded region in Figure 4A) leading to high levels of
plasticity related proteins. After 90 minutes the concentration of proteins
starts to decay back to baseline. While the level of proteins is sufficiently
elevated the consolidation variable zi of each
tagged synapse moves towards zi≈1 since this
is the only stable fixed point of the dynamics (Figure 1C). This leads to a consolidation
time of about 2 hours, enough to switch a large fraction of synapses into the
up-regulated state z≈1 (green line, Figure 4A). Hence the average weight of the
stimulated synapses stabilizes at a value above baseline, indicating L-LTP
(Figure 4A, solid line).

Figure 4. Dynamics of the TagTriC Model during different tagging protocols and
protein synthesis blocking.

The change of the total synaptic weight (top panels, black line ) has contribution from early LTP (top panels, magenta
line represents ) and from late LTP (top panels, green line represents ). The protein variable p (red line,
bottom panels) grows as long as the average number of tags (, blue line) is above the protein synthesis trigger
threshold (Np/N, dashed
horizontal line). For better visibility, the regions where the blue line
is above the trigger threshold is shaded. (A) A strong tetanus
(N = 100 synapses,
stimulated by 100 pulses at 100 Hz, repeated three times every ten
minutes) leads to a sustained period of about 90 minutes where the
number of tagged synapses is above the protein synthesis triggering
threshold (lower panel, blue shaded). During this time the protein
synthesis variable p is close to one (red line, lower
panel), causing an increase in the fraction of consolidated weights
(green line, top panel). (B) During a weak tetanus
(N = 100 synapses,
stimulated by 21 pulses at 100 Hz) the number of tags surpasses the
protein triggering threshold only for a short time which does not enable
switching of the z variable (top panel, green line) to
the up-regulated state. (C) If the weak tetanus is given 30 minutes
after the strong one, the number of tags set by the strong tetanus is
still above the threshold, which allows protein synthesis stabilizing
both the group of 100 synapses receiving the strong tetanus (top panel)
and the group of 100 synapses receiving the weak tetanus (middle panel).
(D) Protein synthesis is blocked for 1 hour (indicated by black bar at
bottom of panel) starting 35 minutes after a first group of 100 synapses
has been stimulated by a strong tetanus. Despite protein synthesis
blocking, both the first group of synapses (top panel) and a second
group of 100 synapses that received a strong tetanus during the blocking
period (middle panel) develop late LTP because proteins synthesized
during the induction of early LTP in the first group decay only slowly
(bottom panel).

If, in a different experiment, 100 synapses are stimulated by the weak tetanus,
the synthesis of plasticity related proteins is only possible during a few
minutes (Figure 4B, red
line), which is not sufficient to switch tagged synapses from
z = 0 into the upregulated
state z≈1. Hence the weights (Figure 4B, black line) decay together with
the tags (Figure 4B, magenta
line) back to baseline and the transition from early to late LTP does not
occur. The decay of the weights is controlled by the rate
kH at which tags stochastically return to zero. The
evolution of the protein concentration p and the consolidation
variable z after a strong tetanus that leads to 90 minutes of
protein synthesis and a weaker tetanus that only leads to 40 minutes of protein
synthesis has been illustrated in (Figure 5A).

(A) Evolution of the variables p and z
during tagging. If protein synthesis is ‘ON’ and the
synapse tagged, p and z move along the
black dashed line towards the stable fixed point on the upper right
(p≈1, z≈1) (red
filled circle). If protein synthesis stops after some time (yellow line,
after 90 min; orange line, after 40 minutes) but the synapse remains
tagged, the dynamics converges towards the fixed point
p = 0,
z = 1 (red filled
circle) indicating that the synapse is consolidated (yellow and orange
trajectories). However, if protein synthesis stops too early (after 25
min, pink line), or if the synaptic tag is lost too early (after 60 min,
magenta line), the synapse is not consolidated and the trajectories
converge towards the non-tagged initial state
p = 0,
z = 0 (red filled
circle). The green dashed vertical line at
z = 0.5 indicates the
threshold beyond which a loss of the tag does not affect consolidation;
the green solid line indicates the separatrix between the stable fixed
points at z = 0 and
z = 1. The minimal
duration of protein synthesis to allow any consolidation is given by the
intersection of the black dashed line with the separatrix. (B) Number of
consolidated synapses (Nup, vertical axis)
as a function of the number of initially tagged synapses
(Ntag, horizontal axis) in simulations
(red filled circles) and theory (solid line). Some of the initially
tagged synapses fail to be consolidated because either they lose their
tag or protein synthesis stops too early (see A). With a protein
synthesis threshold
Np = 40
(arrow) we need about 60 initially tagged synapses to achieve any
consolidation (solid line). If the protein synthesis threshold is
reduced to
Np = 10
(dashed arrow), we need at least 15 tagged synapses to see any
consolidation (dashed line).

The total amount of available protein that is synthesized depends in our model on
the time that the total number of tags stays above the protein triggering
threshold Np. Even though always 100 synapses are
stimulated in our model, not all receive tags in each experiment; moreover
because of the competition for potentiation tags
(hi = 1) and
depression tags
(li = 1) during
induction of plasticity, different synapses can receive different tags in the
same experiment. With our strong tetanus protocol, on average 70 (out of 100)
synapses receive a potentiation tag and 30 a depression tag while with the weak
tetanus the numbers are 30 and 10, respectively. For the depression protocols,
on average 10 synapses receive a potentiation tag and 90 a depression tag under
strong low-frequency stimulation, and typically zero a potentiation tag and 40 a
depression tag under the weak low-frequency protocol. These numbers vary from
one trial to the next so that sometimes the protein trigger threshold
Np = 40 is
reached with the weak protocols and sometimes not. The important aspect is that
even if the threshold is reached for a short time, the duration of protein
synthesis is not long enough to provide a sufficient protein concentration
p for consolidation of the tagged synapses; see Figure 4B and Figure 5A.

Since the concentration p of plasticity related proteins is
crucial for the transition from early to late LTP we wondered how a block of
protein synthesis would interfere with the consolidation of weights in the
TagTriC model. Application of a protein synthesis inhibitor (modeled by setting
the rate kp of protein synthesis to zero) during 1
hour starting thirty minutes before a strong tetanus is given to a group of 100
synapses that would normally lead to L-LTP, induced E-LTP but prevented
consolidation into L-LTP (data not shown). However, if the same simulation
experiment was repeated after a second group of synapses had received a strong
tetanic stimulation 35 minutes prior to the application of protein synthesis
blocker, then both groups of synapses showed consolidation of weights (Figure 4D), consistent with
experiments [12]. Closer inspection of the lower panel in
Figure 4D shows that two
components contribute to consolidation: Firstly, the concentration of plasticity
related proteins (red line) that has increased because of the first strong
tetanic stimulus decreases only slowly back to baseline enabling the switching
of the slow components (variable z, green line) even in the
presence of protein synthesis blocker. Secondly, even after the end of the
application of the blocker, the total number of tags that has been set by LTP
induction is still above the critical value Np
(shaded region in Figure 4D)
so that protein synthesis can be resumed after the end of the blocking period.
In summary, the detailed analysis of the TagTriC model allows to account for
many aspects of tagging experiment in terms of a limited number of
variables.

Discussion

Relation of Models to Experiments

Synaptic plasticity is based on intricate signal transduction chains involving
numerous processing steps and a large number of different molecules [2],[13],[17].
Despite the complexity of the molecular processes, synaptic plasticity has
experimentally been characterized by a small set of distinct phenomena such as
short-term plasticity [44] as well as early and late phases of LTP and
LTD [13].

Existing models of synaptic plasticity have focused on the description of
short-term plasticity [44] and on the induction of LTP and LTD [24]–[26], [33]–[36]. The question of
maintenance has received much less attention and was mainly addressed in the
context of bistability of the CaMKII auto-phosphorylation process [27]–[29], AMPA receptor
aggregation [41], or four identified kinase pathways [45].
While CaMKII is necessary for induction of long-term potentiation [46], it
is probably too narrow to focus modeling studies only on a single or a few
kinases such as CaMKII and neglect other proteins and signaling cascades that
are involved in synaptic maintenance [13]. For example,
there is strong evidence that PKMζ is involved in
synaptic maintenance and necessary for the late phase of LTP in vitro [11]
and in vivo [14]. However, the actual processes are complex
and the molecules involved in setting tags may differ between different parts of
the dendrite. For example PKMζ is involved in setting
tags during E-LTP in the basal dendrite, whereas CaMKII (or MAPK for E-LTD)
plays a similar role in apical dendrites [30].

Instead of focusing on specific signaling cascades, the TagTriC model presented
in this papers aims at describing the essential ingredients of any possible
functional model of L-LTP and tagging. These ingredients include (i) a bistable
switch (described by the dynamics of the
zi-variable) for each synapse that guarantees
long-term stability in the presence of molecular turn-over [16]; (ii) a global
triggering signal for protein synthesis (described by the dynamics of the
p variable); a formalism to (iii) induce early forms of LTP
and LTD and (iv) set synaptic tags. Since we aimed for the simplest possible
model, we have identified the synaptic tags hi and
li for potentiation and depression with the
synaptic weights during the early phase of LTP and LTD, respectively, so that
points (iii) and (iv) are described by the same transition of the synapse from
an initial non-tagged state to the high or low state, respectively. Variants of
the model where the weight during the early phase of LTP and LTD is not directly
proportional to the value of the tags are conceivable.

Even though we do not want to identify the synaptic variables
hi, li,
zi with specific biochemical signals, a
couple of candidate molecules and signaling chains should be mentioned. The
setting of the tag for LTP under normal physiological conditions involves NMDA
receptor activation and elevated levels of calcium which in turn trigger a
signaling chain involving Calmodulin and CaMKII. We therefore think that the
hi variable (representing both the tag for
LTP induction and the weight increase during the early phase of LTP) should be
related to the activation of CaMKII [13],[46].
The molecular interpretation of the tag li for LTD
is less clear [13]. In our model we have taken the tags as
discrete quantities that decay stochastically, but a model with continuous tags
that decrease exponentially gives qualitatively the same results (data not
shown). The reason is that triggering protein synthesis in our model requires a
large number of tags to be set, so that even in the stochastic model only the
mean number of tags is relevant–and the mean
(more precisely, its expectation value) is a continuous variable. Nevertheless,
we prefer the model with discrete values over the continuous one in view of the
switch-like transitions of synapses after induction of LTP and LTD [7],[37]. Maintenance of
enhanced synaptic weights is probably implemented by an increased number of AMPA
receptors in the postsynaptic membrane. Whether the stability arises from a
self-organization process of receptors [41] or from interaction
with persistently activated CaMKII molecules [46] or from additional
kinases such as PKMζ[11],[14], is an open
problem of experimental investigation. Similarly, the exact identity of many
plasticity related proteins is still unknown [13]. In our model we
assume that recently synthesized plasticity related proteins are accessible to
all synapses onto the same postsynaptic neuron. However, a distinction between
proteins synthesized in, say, basal dendrites and that synthesized in apical
dendrites would be possible by replacing the variable p by two
or more distinct variables pk with similar dynamics
(but potentially different trigger thresholds Np),
allowing for a compartmentalization of tagging [13].

Experimental cross-tagging results clearly indicate that there are two different
types of synaptic tags, one for LTP and one for LTD [13],[32],
which we called hi for LTP and
li for LTD, leading to three different states during
tagging (Figure 1A). Since
we have identified the tagging with the early phase of LTP and LTD, our model of
E-LTP and E-LTD also has three different states (whereas our model of late
LTP/LTD has only two states characterized by
zi = 0 and
z− = 1).
The three-state model of early LTP/LTD presented in this paper would predict
that all non-tagged synapses can undergo a transition to E-LTP or E-LTD
depending on the induction protocol–whereas experiments suggest that
about 70 percent of synapses show LTP but not LTD and the remaining 30 percent
LTD but not LTP [7]. Moreover, only those synapses that are
initially weak can be potentiated and only those that are initially strong can
be depressed [7]. This aspect can be included in our model if
we replace the induction rates ρH for LTP by
ρH(1−zi)
and ρL for LTD by
ρlzi so LTP is only
possible from a state with
zi = 0 and LTD only
from an initial state
zi = 1 —
in agreement with a two-state model of early LTP/LTD [7]. For the tagging
and induction experiments presented in this paper, the results do not change
significantly when we implement this extension of the induction model.

Functional Consequences and Predictions

One of the advantages of a simple phenomenological model is that it should be
capable of illustrating the functional consequences of tagging and L-LTP or
L-LTD in a transparent manner. What are these functional consequences?

A characteristic feature that is made transparent in our model (and which we
expect to be present in any model of tagging) is that, under typical
experimental conditions, the transition from early to late LTP is only possible
if a sizable group of synapses have undergone E-LTP or E-LTD. Hence, while
induction of E-LTP is a local Hebbian process that is likely to take place at
the postsynaptic site of the synapse (e.g., the dendritic spine), the transition
from the early to the late phase of LTP requires a minimum number of synapses to
be activated by appropriate stimulation including co-activation of
neuromodulatory input so as to trigger synthesis of plasticity related proteins.
A direct consequence of this is that synapses cannot be considered as
independent. In order to predict whether a synapse memorizes an item for a long
time or forgets it and re-learns some other item, it is not sufficient to
consider a ‘Hebbian’ induction model, where synaptic changes
depend only on the activity of pre- and postsynaptic neurons. For maintenance,
it is not the synapse which decides individually, but it is the neuron as a
whole (or a large functional compartment sharing the same site of synthesis of
plasticity-related proteins [13],[30],[47]) which ‘decides’ whether it
is going to store the present information, or not. Hence, classical [20],[21],[34] and recent [22]
theoretical models which studied memory maintenance in the presence of ongoing
neuronal activity on the level of single synapses need to be
reconsidered, since the assumption of independent synapses does not hold (Figure 5A and 5B). In
particular, our model predicts that, after an ensemble of identical neurons have
received the same stimulus, some neurons learn (adapt a large
fraction of their synapses to the stimulus) and others don't (keep all
their synapses unchanged). With our choice of parameters, this happens in the
TagTriC model if the number of synapses that have been tagged during the
induction protocol is between 55 and 70 (Figure 5B). This neuronal, rather than
synaptic, decision about memorizing an input (see also [48]) is potentially
attractive for prototype learning–a standard paradigm in neuronal
clustering and categorization algorithms, e.g., [19]. In contrast to
traditional neuronal clustering models where learned memories need to be
protected against overwriting by completely different memory items [19],
a model based on tagging would have an intrinsic vigilance threshold via the
trigger threshold Np. Hence it is resistant to
changes at a single synapse.

In our view, the protein synthesis trigger threshold
NP is an important control parameter in the model.
The results of Figure 2F
show that an increase of the trigger threshold reduces the maximal delay after
which a weak tetanus leads to L-LTP after a strong tetanic stimulation in a
different group of synapses. With our normal value of
Np = 40 we need
around 60 synapses to be initially tagged in order to retain any memory. If we
decrease the trigger threshold to
Np = 10 and keep
all other parameters of the model unchanged, then we need at least a group of 15
synapses tagged during the induction protocol to get any consolidation since
some of the initially tagged synapses loose their tag too early to get
consolidated (Figure 5B).
Only for a very small trigger threshold, say
Np = 1, (which
could occur at high concentration of neuromodulators) synapses become (nearly)
independent, since a tag at a single synapse would be sufficient to trigger the
synthesis of proteins which would then become available at that synapse.
Repeated stimulation of the synapse alone would then be sufficient to transform
E-LTP into L-LTP.

In our opinion, the trigger threshold Np is
significantly lower in the presence of neuromodulators such as, for example,
dopamine (for synapses from Schaffer collaterals onto CA1 pyramidal neurons) or
noradrenaline (for synapses in the dentate gyrus). A simple model for the
dependence of Np on dopamine would be
Np = n0/(DAbg+c0)
where n0 is some arbitrary number (say
n0 = 1),
c0 a small number (say 0.001) and DA denotes the
stationary ‘background’ concentration of dopamine (that is,
before the start of the experiment), normalized to
0<DAbg<1. The phasic dopamine signal caused by
co-stimulation of dopaminergic input during tagging experiments is assumed to be
proportional to the number of tags . The trigger condition becomes then equivalent to the condition which shows a trade-off between the phasic dopamine signal and
the stationary background level of dopamine. In particular in the presence of a
large concentration of dopamine (DA≈1), single synapses can be
consolidated. With the assumption that standard tagging experiments in a large
group of synapses are performed at a low dopamine concentration of
DA = 0.024 before stimulation, we retrieve the
value of Np = 40
used in the main part of the results section. The dependence of the trigger
criterion on the number of tags takes implicitly the co-activation of neuromodulatory input
during the experimental stimulation protocol into account: the larger the number
of stimulated neurons and the stronger the stimulus, the higher the probability
of co-activation of dopaminergic fibers. Blocking dopamine receptors amounts in
the model to setting both the background and the phasic dopamine signal to zero.
In this case, protein synthesis is not possible.

Our model of LTP/LTD induction does not only account for voltage and frequency
dependence of LTP/LTD induction, but also for spike timing dependence. In fact,
for a stimulation paradigm where postsynaptic spikes are induced by short
current pulses of large amplitude either a few milliseconds before or after
presynaptic spike arrival, the model of LTP/LTD induction used in the TagTriC
model becomes formally equivalent to a recent model of spike-timing dependent
plasticity [35] which can be seen as an extension of
classical models of STDP [24]–[26]. In the case of
stochastic spiking of pre- and postsynaptic neurons our model shares important
features with the Bienenstock-Cooper-Munro model [33], in particular
the quadratic dependence upon the postsynaptic variables. In addition, our model
also accounts for the voltage dependence of the Artola-Bröcher-Singer
model [38]. Thus, the model of LTP/LTD induction shares
features with numerous established theoretical models and covers a large range
of experimental paradigms known to induce LTP or LTD [3]–[6],[8].

Since the subsequent steps of protein synthesis trigger and stabilization are
independent of the way early phase of LTP is induced, our model predicts that
tagging experiments repeated with different stimulation paradigms, but otherwise
identical experimental preparation and age of animal, should give similar
results as standard tagging protocols. In particular we propose to stimulate a
group of synapses in hippocampal slices by 40–60 extracellular current
pulses at 10 Hz while the postsynaptic neuron is receiving intracellular current
injection that triggers action potential firing either a few milliseconds before
or after presynaptic spike arrival and keeps the membrane potential at a
depolarized level between postsynaptic action potential firing. Our model
predicts that this will induce early LTD or LTP depending on spike timing and
depolarization level that is not maintained beyond 1 or 2 hours. However, if the
same stimulation occurs after a second group of synapses has received a strong
tetanus, then stabilization of synapses at potentiated or depressed levels
should occur, similar to standard tagging and cross-tagging experiments. In our
opinion, these predictions should not depend on model details, but hold for a
broad class of models that combine a mathematical description of induction of
synaptic plasticity with a mechanism of consolidation.

Another finding—which is somewhat unexpected and in contrast to other
conceptual models of synaptic tagging and capture [12],[13],[47]—is that during a strong tetanic
stimulation a fraction of synapses receives tags for depression (while most, but
not all, receive tags for potentiation). This is due to the fact that during
induction of plasticity, transition to E-LTP and E-LTD act in parallel [7].
The prediction is that after consolidation (say 2 hours after the strong tetanic
stimulation) a small fraction of synapses would show L-LTD, rather than L-LTP.

An essential ingredient of our model that allows long-term stability of
consolidated synapses is the bistable dynamics of the variable
z. In our opinion, such bistability (or possibly multistability
[49] with three or four stable states) is necessary
for synaptic maintenance in the presence of molecular turn-over, as recognized
in earlier theoretical work [15],[16],[34]. Our
model therefore predicts that L-LTP and L-LTD should have bistable, switch-like
properties. While there is evidence for switch like transitions during the
induction of E-LTP and E-LTD [7],[37], the bistability
of the late phase of synaptic plasticity has so far not been shown. A possible
experiment would be to combine a minimal stimulation protocol (e.g., a weak
tetanus) at a single synapse [7],[37] with a medium to
strong stimulus at a group of other synapses (e.g., tetanic stimulus varying
between 30 and 100 pulses). The prediction is that the weight of the single
synapse shows an all-or-none phenomenon with transition probabilities that
depend on the stimulation of the group of other synapses. In particular, as the
number of pulses of the tetanic stimulation is reduced (covering a continuum
from strong to weak tetanic stimulation), the maintenance in the potentiated
state should become less likely (averages across many experiments decrease)
whereas the results of individual experiments show either full potentiation or
none, which should give rise to a bimodal distribution of normalized synaptic
weights.

Open Questions and Perspectives

A lot of questions remain open and need to be addressed in future studies. First,
can a synapse that has been potentiated in the past and is maintained after a
transition to late LTP undergo a further potentiation step [13]? In our current
model this is not possible since the consolidation variable z
has only two stable fixed points. If we replace the function
f(z) depicted in Figure 1 by another one with more than two
stable fixed points, then the answer to the above question would be positive.
Indeed, there have been suggestions that self-organization of receptors into
stable sub-groups could lead to multiple stable states [49].

Second, induction of LTP or LTD is not only possible by strong extracellular
stimulation of groups of synapses, but also at single synapses if presynaptic
activity is paired with either a depolarization of the postsynaptic membrane
[5],[7] or tightly timed
postsynaptic spikes as in STDP experiments [6],[8]. How can
it be that the change induced by STDP seems to be maintained over one hour
without visible degradation? [6],[7]. Are synapses in
these experiments consolidated, and if so what is the concentration of
neuromodulators? In the TagTriC model with the choice of parameters used in the
present paper, consolidation would not be possible, since the minimum number of
synapses that have undergone E-LTP or LTD is
Np = 40 in order to
trigger protein synthesis, but, as explained above, an increased neuromodulator
concentration would make consolidation possible.

Third, what is the role of NMDA receptor activation during synaptic
consolidation? In our present model, protein synthesis is triggered by
appropriate induction protocols, but is independent of synaptic activity during
the consolidation process. However, recent experimental results suggest that
protein synthesis blocker needs synaptic stimulation during the consolidation
period to become effective [50], suggesting a subtle interplay between
protein synthesis and synaptic activation that cannot be captured by our model.

Fourth, has each neuron a single protein synthesis unit or is protein synthesis a
local process confined to each dendritic branch? In the first case, there is a
single neuron-wide protein synthesis trigger threshold [12] and the neuron as a
whole ‘decides’ whether early forms of synaptic potentiation
and depression will be consolidated or not. This is the paradigm posited in the
TagTriC model. In the alternative model of local protein synthesis [13],[47], the critical
unit for consolidation are local groups of synapses on the same dendritic
branch. Thus, for the same number of tagged synapses, a local group of synapses
on the same dendritic branch is more likely to undergo consolidation than a
distributed set of tagged synapses, leading to a form of clustered plasticity
[47]. The TagTriC model can be easily adapted to
the case of clustered plasticity by (i) replacing the point-neuron model by a
neuron model with spatially distributed synapses and (ii) replacing the
neuron-wide trigger equation (see 4 and Figure 1B) by a finite number of analogous,
but dendrite-specific equations.

Fifth, how can tags be reset? Experiments show that a depotentiating stimulus
given 5 minutes after a weak tetanus erases the trace of E-LTP (resets the tag)
whereas depotentiation 10 or 15 minutes after the strong tetanus only
transiently suppresses the E-LTP, making the consolidation of the synapse by
protein capture possible [51]. We have checked in additional simulations
that our present model cannot account for these experiments. In our opinion, the
above tag-reset experiments show that the synapse has additional hidden states
currently not included in the TagTriC model. Additional states would allow to
(i) separate the measured early LTP during the first 5 minutes from setting the
tag; and (ii) distinguish between depotentiation and depression of synapses. One
interpretation of the tag-reset experiments [51] is that during
the first five minutes the tag is not yet set whereas early LTP is already
visible. The tag would be set only with a delay of 5–10 minutes.
Application of a depotentiating stimulus more than 10 minutes later would then
leave the potentiation tag intact, but move the synapse to a transiently
depotentiated state.

The final and potentially most interesting question is that of functional
relevance: Can the TagTriC model be used to simulate reward-based learning in
experiments in vivo [13]? The formal theory of reinforcement learning
makes use of an eligibility trace [52] which can be
interpreted as a synapse specific tag. In the future we want to check whether
the TagTriC model can be linked to reinforcement learning models [53]–[56] under the
assumption that reward prediction errors are represented by a dopamine signal
[57] which influences the protein synthesis dynamics
in our model. This open link to reward-based learning is of fundamental
functional importance.

Methods

Model of Early LTP/LTD and Tagging

In our model we assume that presynaptic spike arrival needs to be combined with a
depolarization of the postsynaptic membrane (e.g., [5]) in order to induce
a change of the synapse. In voltage clamp experiments (e.g., [39])
the postsynaptic voltage would be constant. However, in general the voltage is
time-dependent and described by a variable
u(t). In the TagTriC model, we assume that the
low-pass-filtered voltageneeds to be above a critical value
ϑLTD to make a change of the synapse
possible. τlowP is the time constant of the
low-pass filter and
ϵ = 1 ms is a short
delay twice the width of a spike (see Table 1). This short delay ensures that
u̅ includes effects of previous presynaptic
inputs and postsynaptic spikes, but not of an ongoing postsynaptic action
potential.

Table 1. Parameter values used throughout all simulations, except Figure 1E–G
where
Np = 10 and
initial percentage of
zi = 1 was
10%, because these simulations refer to experiments with
younger animals.

Combining presynaptic spike arrival at synapse i (represented by
xi) with a depolarization
u̅ of the postsynaptic neuron above a threshold
ϑLTD we get a rate of LTD(1)where ALTD>0 is a parameter
and [.]+ denotes rectification, i.e.,
[y]+ = y
if y>0 and zero otherwise. Here denotes the presynaptic spike train with pulses at time and δ the Dirac-delta function.
Formally, ρL describes the rate of
stochastic transitions from the non-tagged state
h = 0,
l = 0 to the low state
l = 1, Figure 1. In simulations we
work with discrete time steps of Δ = 1
ms. Eq. 1 indicates that the probability
Pl = 0→l = 1
of a transition to the low-state during the time step Δ vanishes in the
absence of presynaptic spike arrival and takes a value of
Pl = 0→l = 1 = 1−exp(−ALTD[u̅(t)−ϑLTD]+Δ)≈ALTD[u̅(t)−ϑLTD]+Δ
if a presynaptic spike arrives at the synapse i during the time
step Δ. Note that the transition from
l = 0 to
l = 1 is only possible if
h = 0 and
h remains zero during the transition.

Similarly, a switch from the non-tagged state
h = 0,
l = 0 to the high state
h = 1 occurs at a rate
ρH which also depends on
postsynaptic voltage and presynaptic spike arrival. We assume that each
presynaptic spike at synapse i leaves a trace
x̅i that decays exponentially with time constant
τx. The exact biophysical nature of the
trace is irrelevant, but could, for example, represent the amount of glutamate
bound to the postsynaptic receptor. The value of the trace at time
t caused by earlier spike arrivals at time is then where the sum runs over all firing times . With the trace x̅i we write(2)which indicates that, in addition to the conditions for LTD
induction we also require the momentary membrane potential
u(t) to be above a second threshold
ϑLTP. This threshold could change on
the time scale of minutes or hours as a function of homeostatic processes. To
summarize, the rate of LTP transition ρH is
different from ρL in five aspects. First,
the constant ALTP is not the same as
ALTD. Second, LTP is caused by the trace
x̅i left by presynaptic spikes, rather than the spikes themselves. This
trace-formulation ensures that presynaptic spikes can interact with later
postsynaptic spikes as in classical models of STDP [24]–[26].
Third, the time constant of the low-pass filter in u̅
is different; fourth, the momentary voltage needs to be above a threshold
ϑLTP; and fifth, the total dependence
upon the postsynaptic voltage is quadratic, rather than linear. The quadratic
dependence ensures that for large depolarization LTP dominates over LTD [39].
Tagged synapses with
hi = 1 decay with
probability
Ph = 1→h = 0 = kHΔ
back to the non-tagged state (and analogously, but with rate
kL for the transition
li = 1→li = 0).

In the TagTriC model, the local synaptic values
h = 1 for potentiation or
l = 1 for depression act
as tags indicating potential sites for further consolidation, but are also
directly proportional to the weight of the synapse after induction of LTP or
LTD. Since in minimal stimulation experiments LTD leads to a reduction of about
50 percent of the synaptic efficacy whereas LTP leads to an increase by up to
100 percent [7], we model the weight change during the early
phase of LTP as
Δwi = (hi−αli)w̅
where w̅ is the weight of the non-tagged synapse and
α = 0.5. The total
weight change Δw/w̅ measured
shortly after induction of LTP or LTD with extracellular protocols corresponds
to the fraction of synapses in the high or low states, respectively, hence, if
all synapses start from the non-tagged state the measured weight change is where N is the number of synapses stimulated
by the protocol. The set of parameters of LTP/LTD induction and tagging is given
in table 1.

Trigger

The triggering process is controlled by the dynamics of a variable
p which describes the amount of plasticity related proteins
synthesized in the postsynaptic neuron. Protein synthesis is triggered and the
variable p increases while the concentration of dopamine
exceeds a critical level ϑp[58]. If the dopamine concentration DA falls below
ϑp, the protein concentration
decays with a time constant τp. Assuming
standard first-order kinetics we have(3)Protein synthesis has a maximum rate
dp/dt of kp
and saturates if the amount of protein approaches a value one.
Θ[y] denotes the unit step
function with
Θ[y] = 1
for y>0 and zero otherwise.

Dopamine is present at a low stationary background value. In addition a phasic
dopamine component is induced in standard tagging experiments in hippocampal
slices, because of co-stimulation of dopaminergic inputs during extracellular
stimulation of presynaptic fibers [40]. To describe the time course of the phasic
dopamine component in our model, we assume that the dopamine is proportional to
the total number of tags Σi(hi+li)
induced by the stimulation protocol. The stationary background level of dopamine
DAbg is included in the threshold
ϑp = Np(DAbg)
for protein synthesis. Hence Eq. 3 can be rewritten in the form(4)Note that we have chosen units so that the threshold for protein
synthesis Np can be interpreted as the minimal
number of tags necessary to stimulate protein synthesis. This interpretation is
important for the discussion of the model results, in particular Figures 4 and 5.

A suitable model for dependence of the protein synthesis threshold on the
background level of dopamine is
Np(DAbg) = n0/(DAbg+c0)
where n0 = 1 is a
scaling factor,
c0 = 0.001 a
constant and 0≤DAbg≤1 is the normalized dopamine
concentration. We note that the trigger condition [Σi(hi+li)−Np(DAbg)]>0
is then equivalent to the condition (DAbg+0.001)[Σi(hi+li)]>1.
This formulation shows that there is a trade-off between background levels and
phasic dopamine. Unless stated otherwise we always use in the simulation a fixed
dopamine level DAbg = 0.024 so that
Np = 40.
The specific model Np(DAbg) of the
dependence upon background dopamine levels is therefore irrelevant.

We assume that the plasticity related protein p synthesized in
the postsynaptic neuron is diffused in the dendrite of the postsynaptic neuron
and hence available to all the synapses under consideration. Hence, the tags
hi and li have
indices, since they are synapse-specific, whereas p in Eq. 4
does not.

Consolidation and Late LTP

The consolidation variable z describes the late phase of LTP and
follows the dynamics(5)

The scaling factor γ is a function of the dopamine level
DA. In the simulations we always assumed a fixed dopamine
level and set
γ(DA) = 0.1.

In the absence of plasticity related proteins
(p = 0), or if no tags are set
(hi = li = 0),
the function
f(z) = z(1−z)(z−0.5)
generates a bistable dynamics with stable fixed points at
z = 0 and
z = 1 and an unstable fixed
point at z = 0.5 marked by the
zero crossings of the function f, Figure 1C. In the presence of a finite amount
of proteins p>0 and a non-zero tag, the location of the
fixed points changes and for p>0.47, only one of the
stable fixed points remains. The potential shown in Figure 1C is a function E
with
dE/dz = −f(z)
so that
dz/dt = −dE/dz.
We note that a synapse i can change its consolidated value only
if both a tag
(hi = 1 or
li = 1) and
protein p>0.47 is present–summarizing the
essence of ‘synaptic tagging and capture’ [12],[13].

Synaptic Weight

The synaptic weights have contributions from early and late LTP and LTD. The
total synaptic weight of a synapse i is
wi = w̅(1+hi−αli+βzi)
where w̅ is the value of a non-tagged synapse,
α = 0.5 and
β = 2 are
parameters, hi and li
are binary values indicating E-LTP and E-LTD, respectively, and
zi is the value of the L-LTP trace of synapse
i. Since we model slice experiments in animals older than
20 days, we assume that 30 percent of the synapses have undergone previous
potentiation and have z = 1
while the remaining 70 percent of synapses are in the state
z = 0 [7]. In all simulation
experiments we stimulate one or several groups of
N = 100 synapses each.
Assuming that no tags have been set in the recent past
(hi = li = 0),
the initial value of the average weight in a group of N
synapses is then .

Neuron Model

For all simulations in this paper we use the adaptive exponential
integrate-and-fire model [42] as a compact description of neuronal firing
dynamics. Briefly, it consists of two equations. The voltage equation has an
exponential and a linear term as measured in experiments [59]. The second equation
describes adaptation. Although firing rate adaptation is not important for the
present study, it would be relevant in the context of other stimulation
paradigms. Parameters for the neuron model are as in [42] and are kept fixed
for all simulations presented in this paper. The voltage threshold
Vs of spike initiation by a short current
pulse is 25 mV above the resting potential of −70.6 mV [42].
Synaptic input is simulated as a short current pulse. The initial connection
weight w̅ was adjusted so that simultaneous
activation of 40 or more synapses triggers spike firing in the postsynaptic
neuron. Hence the amplitude of a single EPSP is about 0.6 mV.

The adaptive exponential integrate-and-fire model is defined in continuous time.
If a spike is triggered by a strong current pulse, the voltage rises within less
than 0.5 millisecond to a value of 20 mV where integration is stopped. The
voltage is then reset to resting level, and integration restarted after a
refractory time of 1 ms. In order to enable us to perform simulations of
plasticity experiments with a time step of
Δ = 1 ms, the voltage equation during
the rising slope of the action potential was integrated once at a much higher
resolution (time step 0.02 ms), so as to determine the exact contribution of
each postsynaptic spike to the probability of LTP induction. Every postsynaptic
spike was then treated as an event in the plasticity simulations that
contributed a probability
Ph = 0→h = 1
of flipping the tag from h = 0
to h = 1 in a time step
Δ = 1 ms which we can write as
Ph = 0→h = 1 = aΔx̅(t)[u̅(t)−ϑLTD]+
with a numerical conversion factor
aΔ = ALTP
5 ms mV derived by the above procedure; see Eq. 2.

Number of Consolidated Synapses

In Figure 5 we plot the
number of synapses that have been consolidated as a function of the number
Ntag of initially tagged
(hi = 1) synapses.
Since the number of tags decays exponentially with rate
kH, the expected duration of protein synthesis is where Np is the protein trigger
threshold. While protein synthesis is ‘ON’ the variables
p and z move along the black dashed line
in Figure 5A which crosses
after a time t1 the separatrix (green line in Figure 5A) and at a time
t2 the line
z = 0.5 (vertical dashed green
line). Different cases have to be distinguished. (i) , no consolidation takes place (see pink trajectory), hence
Nup = 0. (ii) , consolidation is guaranteed for all synapses that are still
tagged at time t2, hence
Nup = Ntagexp(−kt2).
(iii) In the case of , the time tcross needed to cross
the vertical line z = 0.5 is
numerically calculated by integrating the equations
dp/dt = −p/(τp)
and
dz/dt = f(z)+γ
p starting at at the point on the black-dashed line (see orange line in Figure 5A for a sample
trajectory). The number of consolidated synapses is then
Nup = Ntagexp(−ktcross).
The solid line in Figure 5B
represents Nup as a function of
Ntag calculated for the cases
(i)–(iii). With our standard set of parameters, we have
t1≈28 min and
t2≈60 min.

Acknowledgments

We thank Julietta Frey and Mark van Rossum for helpful discussions.

Author Contributions

Conceived and designed the experiments: CC WG. Performed the experiments: CC.
Analyzed the data: CC. Wrote the paper: WG. Designed the model of late LTP
and performed research: LZ. Participated in research on a precursor model of
early LTP/LTD: EV LB.